Error-rate Estimation with Ones Counting

Transcription

1 UC TCHNICA RPORT CNG-8- ng Hseh epatment of lectcal ngneeng teb chool of ngneeng Unvesty of ouen Calfona o-ate stmaton w Ones Countng Zhaolang Pan elvn A Beue Abstact W I ccut featue sze scalng down, t s becomng moe dffcult expensve to acheve a desed level of yeld o-toleance, at s, employng defectve chps at occasonally poduce eoneous yet acceptable esults n tageted applcatons, has been poposed as one way to ncease effectve yeld In e doman of eo-toleance, defectve chps ae chaactezed by cetan ctea set by vaous applcatons o-ate, namely how fequent eos occu at e output, s one such cteon In s epot we focus on e followng poblem: Gven a combnatonal logc ccut at s defectve hence occasonally poduces an eoneous output, how can one detemne e eo-ate of each output lne by usng ones countng? Ths poblem was pevously consdeed by oes but e analyss has some naccuaces Ths epot pesents bo a moe accuate analyss new estmato of eo-ate C I INTROUCTION lasscal I post-manufactung tests patton chps nto two categoes, namely ose at pass ose at fal Pesumably, chps at fal have defects ae dscaded, whle e oes ae sold to customes In s epot we wll use e tems fal defectve chp ntechangeably The facton of defect-fee chps among all manufactued chps s called e yeld of a I manufactung pocess W mnmal featue szes scalng down to below 9nm, t s becomng moe dffcult, hence expensve, to acheve a desed level of yeld [] efectve chps contbute to a lage economc loss to chp manufactues, nceased pces to consumes In an attempt to ncease yeld, desgnes employ seveal technques such as fault-toleance, defect-toleance desgn-fo-manufactuablty [] [] [] Anoe somewhat adcal new technque oogonal to ese s called eotoleance [5] [6] A chp s sad to be eo-toleant T w espect to an applcaton o system f t contans defects hence occasonally poduces eos, usng s chp n a gven applcaton/system poduces acceptable esults to e Ths epot s based upon wok suppoted n pat by e Natonal cence Foundaton unde Gant No 89 end use Because a facton of defectve chps ae used n systems ae an beng dscaded, eo-toleance nceases effectve yeld, hence potentally poft any examples of systems at can opeate n an eotoleant mode esde n e doman of mult-meda manmachne ntefaces ome smple applcatons ae befly descbed below Whle e applcaton of eo-toleance n dgtal systems pobably dates back seveal decades, one of e ealest documented applcatons we have encounteed appeas n a 995 patent: It s heen ecognzed at n an audo sgnal pocessng system, a RA chp fo stong dgtzed audo sgnals selected to nclude at least one nopeatve memoy locaton, s acceptable fo use as a stoage medum n at no notceable eo s poduced on playback of e ecoded sgnal due to e samplng ate of e audo sgnal due to e elatvely low ate of defects allowed Fuemoe, e use of a lessan-pefect RA chp fo stong audo nfomaton s acceptable due to e substantally non-ctcal natue of audo sgnals, as opposed to e extemely ctcal natue of compute data [7] The followng statement appeas n anoe patent dealng w a telephone-answeng machne: The use of audo RAs ARAs s also known n e answeng machne at, w ARAs beng RAs at ae allowed to have a small numbe of defectve bts, n ode to allow e use of lowecost ntegated ccut memoy chps [8] Bo statements embody e concept of eo toleance, e, at some defectve ccuty can poduce acceptable pefomance An PG ccut s anoe example whee eo-toleance s applcable The effect of defects n some functonal blocks of a PG encode, such as e moton estmaton block, on e pefomance of e ccuts has been studed [9] [] The esults ndcate at fo about 5% of ose defects at can be modeled as a sngle o double stuck-at fault, e esultng defectve PG ccuts geneate acceptable pefomance Oe examples of eo-toleant applcatons have been epoted elsewhee [] [5] o-toleant chps ange fom low pced memoes to hgh pced audo vdeo pocessng chps In e nea futue, many system-on-chp

2 UC TCHNICA RPORT CNG-8- poducts wll be excellent cdates fo eo-toleant applcatons, such as ose contanng embedded language tanslatos as well as audo, touch, smell /o vdeo pocessos Gven an analog system, one can nject eos nto e ccut at some node by applyng nose at at node measue attbutes of e output, such as db loss, O value, o nstablty In a dgtal ccut, one can ntoduce eos on an ntenal sgnal lne measue e esultng mpact on e data at e output bus Two measues at have been used to quantfy output eos ae eo-ate eo-sgnfcance [5] [6] A fomal defnton of eo-ate s povded n [] o-ate ndcates how often on aveage an eo s seen at e output when om nput pattens ae appled If e ccut s combnatonal contans a defect, e pattens ae unfomly selected ove e nput space, en e eo-ate s equvalent to e facton of e nput space at detects e defect, e, poduces an eoneous output o-sgnfcance ndcates e magntude of an eo, when e assocated patten epesents some numec quantty ependng on e applcaton ee one o bo of ese measues mght be applcable R C R Fg mplfed BIT achtectue In s epot we focus on estmatng e eo-ate of a defectve ccut havng statc defects Based on e eoate, defectve chps can be bnned nto categoes Fo example, two eo-ate esholds, say, whee < < <, can be used to dvde e ange of eo-ate nto ee pats, e,, ], < ], ] efectve chps can now be pattoned nto ee categoes, namely categoy I w eo-ate n e ange, ], categoy II n < ], categoy III n, ] Chps n categoy III hgh eo-ate ae dscaded; chps n categoy II ae sold at a lage dscount n pce; ose n categoy I ae sold at a modeate dscount Ou technque fo estmatng eo-ate eques at we apply a lage numbe of pseudo-om pattens to a tageted ccut ost complex moden ccuts employ ee a fullscan FT o a BIT meodology [6] [7] Whle ee of ese test meodologes can be used to mplement ou eoate estmaton technque, we wll descbe ou wok n e context of a BIT meodology snce s smplfes e dscusson A smplfed veson of a BIT achtectue used O efes to mean opnon scoe, ndcates e qualty of audo speech afte pocessng by some COC to test combnatonal logc s shown n Fg In nomal opeaton, egste R eceves data n paallel dves e block of logc C C n tun passes data to R In e desgnfo-test pocess, ese egstes ae modfed to opeate dffeently when n e test mode In e test mode, R opeates as a pseudo-om patten geneato PRPG, R as a multple nput sgnatue analyze IR The opeaton of e test pocess s as follows Fst e egstes ae ntalzed to some known states Then e egstes ae clocked tmes, whee s often a vey lage numbe At e end of s pocess, f e state of e IR s e same as some pe-computed value, en e ccut C s sad to pass s good o defect fee, oewse t fals The dea hee s at f ee s a defect n e ccut, fo at least one of e nput pattens e coespondng output patten would be n eo, once an eo entes e IR, w a vey hgh pobablty e state of e IR emans n eo even as new data aves at ts nputs W some assumptons egadng e dstbuton of output pattens, s technque can be effectvely appled to some foms of sequental ccuts, such as feed-fowad ppelnes Thus, to mplement e bnnng pocess, t s mpotant to effcently bound e eo-ate of a defectve chp Note at e test meodology dscussed n s epot does not employ a fault model, test patten geneaton o fault smulaton A technque fo estmatng e eo-ate of a block of logc usng sgnatue analyss was fst poposed n [] ubsequently, Pan Beue sgnfcantly exped on e eoetcal aspects of s meodology [] hahd Gupta en poposed a vaant of s wok usng ones countng ae an sgnatue analyss [] Unfotunately, e analyss mpled at eo-ate estmaton va ones countng used sgnfcantly moe esouces an at equed va sgnatue analyss As wll be explaned late, e esults ae counte ntutve In s epot, we befly evew e analyss pesented n [], pesent a new analyss fo s ones countng technque a new estmato of eo-ate The esults fom ou analyss show at ones countng sgnatue analyss technques use compaable esouces fo obtanng e same degee of accuacy n eo-ate estmaton Ths epot s oganzed as follows ecton evews e ones countng eo-ate estmaton wok pesented n [] In ecton, we pesent ou new analyss of s technque A new estmato s poposed e statstcal chaactestcs of e estmato ae pesented eveal specal cases of usng e ones countng technque ae descbed n ecton In ecton 5, smulaton esults ae pesented The poblem of chp classfcaton based on ones countng eo-ate estmaton s dscussed n ecton 6 ecton 7 pesents e expemental esults of chp classfcaton, ecton 8 concludes e epot II RIW OF PRIOU WORK The eo-ate estmaton technques poposed n [5], [6],

3 UC TCHNICA RPORT CNG-8- [] [] ae llustated usng a BIT achtectue, ough a scan achtectue s equally applcable The egste dvng e ccut-unde-test CUT s a PRPG, whch s usually mplemented va a lnea feedback shft egste FR In [], e output of e CUT s compessed usng anoe FR, whle n [] a nonlnea fnte state machne s used to obtan e ones countng n e output sequence When bo ccuts opeate n e test mode, a sequence of test pattens ae geneated appled to e CUT Fo e ones countng meodology, e ones count n e output sequence s ecoded en compaed w e pe-computed ones count of e fault-fee ccut usng e same nput pattens The dffeence,, s stoed Fo e sgnatue analyss technque, e fnal sgnatue s compaed to e pe-computed coect sgnatue If ey ae e same e test s sad to pass; oewse t fals Ths pocess of applyng pattens detemnng e value of, o pass/fal, s efeed to as a test sesson The pocess fo estmatng eo-ate conssts of cayng out test sessons, each usng a dffeent subset of test pattens fom e space of all test pattens In e followng, we wll concentate on e ones countng technque The analogous yet adcally dffeent analyss fo sgnatue analyss s gven n [] Cayng out ese test sessons esults n dffeences, denoted by,,, s It s assumed at all test pattens ae omly geneated at e total numbe of pattens, namely x, s a small facton of e nput patten space Thus ese numbes ae ndependent have e same dstbuton The sample mean sample vaance ae gven by e equatons Based on e sample mean vaance, e eo-ate estmato, ˆ, gven n [] s ˆ N N whee N n, n s e numbe of nput pns of e CUT Agan, N>> The selecton of depends on what objectve s to be satsfed In [], e auos analyzed e poblem of usng s ones countng technque to classfy chps accodng to e estmated eo-ate et epesent an eo-ate eshold value, usually specfed by an applcaton, ε a magn of eo allowed n e value of an estmated eo-ate, γ e confdence assocated w s estmaton If e estmated eo-ate, ˆ, of a CUT s smalle an, e CUT s accepted, oewse t s ejected The followng equements ae set fo e estmaton of eo-ate Fo a CUT w actual eo-ate, a If < ε, e pobablty of acceptng e CUT s geate an o equal to γ That s Pob ˆ < γ f < ε b If > ε, e pobablty of ejectng e CUT s geate an o equal to γ That s Pob ˆ > γ f > ε c If ε ε, e CUT may be accepted o ejected In [] e auos dscussed lowe bounds fo such at e above equements ae satsfed The analyss showed at e lowe bound fo e numbe of test pattens pe test sesson,, s, e lowe bound γ ε fo e numbe of test sessons,, s γ ε ε / ε Assume, ε γ977 The lowe bound of s about 97 e lowe bound of s about 7 Because e ones countng technque eques knowledge of e ones countng fo e fault-fee ccut n ode to compute, values must be stoed, whch esults n an exobtant amount of ovehead mployng e sgnatue analyss technque fo e same equements as above, only test pattens ae needed fo each of 58 test sessons These esults ae counte-ntutve fom e followng pont of vew In sgnatue analyss one only gets a bnay decson fom each test sesson, namely pass o fal Ignong e ssue of alasng, s decson s always coect, but e fal can be caused by,, up to eoneous esponses Hence a geat deal of nfomaton s lost pe falng test sesson On e oe h, fo ones countng, one detemnes an actual obseved ones countng, O, of,, *, Then e ecoded value s O O, at can take * on e values,,,,,,,,,, whee O s e ones countng fo e fault-fee ccut Hence ee appeas to be a much lage amount of nfomaton n an n pass/fal In fact, f, en fo sue e test has faled But f, e test may have passed o faled Thus alasng plays a bg ole n ones countng Howeve, t seems lke e value of gettng a non-bnay esult should moe an compensate fo e poblem of alasng In addton, f en no alasng s possble, so f e esponse s coect, f t s ncoect Hence, fo a lage enough value of, e eo-ate can be accuately estmated, us e lowe bound on s Thus t s useful to eexamne e analyss of eo-ate estmaton va ones countng To moe fully chaacteze e statstcal popetes of e estmato ˆ, ts mean vaance should be computed The mean of e estmato wll be close to e tue eo-ate when e estmato s based, equal to e tue eo-ate when e estmato s unbased The vaance of e estmato s not dectly calculated n [] Instead, t s mpled to be equal to e vaance of anoe om vaable namely e n [], whch s not always tue In s epot, we pesent an analyss of e ones countng technque at s dffeent fom e analyss povded n [] Ou analyss moe accuately descbes e chaactestcs of

4 UC TCHNICA RPORT CNG-8- ones countng technque fo e pupose of eo-ate estmaton We also compae s ones countng technque w e pevously publshed wok usng sgnatue analyss The compason shows at s ones countng technque s able to estmate eo-ate as effectvely as e sgnatue analyss technque, e hadwae oveheads fo bo technques ae compaable III TATITICA ANAI Consde a sngle output combnatonal ccut, C, a faulty veson of s ccut, C f In esponse to each nput patten, e output of C f can be classfed nto one of fou types, namely, /, /, / /, whee / means at e output of bo C C f ae ; / means at e output of bo C C f ae ; / means e at e output of C s C f s ; / means at e output of C s C f s If all possble N nput pattens ae appled to C f, a sequence of N outputs consstng of e fou types of outputs s geneated The eo-ate of C f s e ato of e numbe of outputs of type / plus type / to N ung a sngle test sesson, e obseved ones count s equal to e numbe of type / / outputs type x/ The esult s subtacted by e coect ones countng of C, whch s equal to e sum of type / / outputs of C f Ths dffeence, denoted by, epesents e dffeence between e ones count geneated by C f e ones count geneated by C It also equals e numbe of type / outputs mnus e numbe of type / outputs of C f A complete test conssts of test sessons Thus, ecoded ae numbes, each of whch s e value of fo a test sesson The eo-ate s estmated accodng to ese numbes Imagne at N possble outputs defne a collecton havng fou types of symbols In each test sesson, we choose outputs wout eplacement fom e collecton Fom e selected outputs, we do not ascetan e numbe of type / o / outputs, but e dffeence between e numbe of / / outputs Afte a test sesson s fnshed, we put ese outputs back nto e collecton test sessons ae caed out, esultng n numbes Fom ese numbes, we estmate e facton of outputs n e collecton at ae ee of type / o / et p be e facton of / outputs n e collecton, p e facton of / outputs n e collecton Thus, p p p, p ae all postve less an An oacle knows p p We wsh to estmate p p snce e estmated eo-ate equals e sum of e estmated values of p p Assume e outputs ae dawn one at a tme, we have a counte, ntalzed to If a type / output s dawn, e value of e counte s nceased by ; f a type / output s dawn, e value of e counte s deceased by ; f a type / o / output s dawn, e state of e counte s not changed The fnal state afte outputs ae chosen, e, a test sesson, s just e state of e counte, say Because outputs ae dawn omly, s a om vaable In e above pocess, t s mpled at we ae samplng wout eplacement Because N s assumed to be lage w espect to, whch s usually e case n pactce, e change n e facton of each type of output n e emanng collecton afte each of e outputs s selected s vey small can be gnoed o we wll teat s pocess as samplng w eplacement Thus, fo each dawng, e pobablty at e counte nceases by s p, e facton of / outputs; e pobablty at t deceases by s p, e facton of / output; e pobablty at t does not change s p p et X be a om vaable such at X w pobablty p, X w pobablty p, X w pobablty p p Thus X X X, whee X, X,, X ae dentcally ndependently dstbuted d om vaables w e same dstbuton as X Fom e pobablty densty functon PF of X, we see at xpectaton: { X p p aance: a{ X p p p p Thus e expectaton vaance of ae xpectaton: { { X X X { X p p aance: a { a{ X X X a{ X p p p p Fom e test pocess we obtan samples of e om vaable, namely,,, The sample mean e sample vaance ae defned by e equatons We ntend to estmate e two paametes, p p, n e dstbuton of A genec meod fo buldng an estmato s based on appoxmaton of e moments of e om vaable [] The fst ode moment of a om vaable s ts expectaton, whch s appoxmated by e sample mean The second ode moment s ts vaance, whch s appoxmated by e sample vaance Thus, we have { p p p a{ p p p p 5 Fom 5 we solve fo p p, obtan, hence p p Because e eo-ate s equal to e sum of p p, e estmated eo-ate, ˆ, s p

5 UC TCHNICA RPORT CNG-8-5 ˆ, 6 whch s also called e estmato of eo-ate ˆ s a functon of samples of o e estmato tself s a om vaable To evaluate e pefomance of an estmato, ts expectaton vaance need to be computed If e expectaton s equal to e tue value of e estmated quantty, e estmato s unbased; oewse t s based If e estmato s based, e dffeence between e expectaton of e estmato e tue value of e estmated quantty s of nteest malle dffeences mply bette estmatos The vaance of e estmato epesents how close e estmaton s to e expectaton of e estmato A lage vaance means e PF s somewhat flat e estmaton esult s lkely to be poo A small vaance mples at e PF s naow aound ts expectaton, e estmaton esult s lkely to be close to e expectaton of e estmato W e expectaton vaance of e estmato, we ae able to appoxmate e PF of e estmato If e type of dstbuton of e estmato s known, e PF of e estmato can be expessed explctly If e type of dstbuton s unknown, as t s fo e eo-ate estmato, e PF of e estmato s usually appoxmated by a nomal dstbuton, whch s a functon of e expectaton vaance of e estmato The pocedue fo devng e expectaton e vaance of e estmato ˆ s gven n Appendx A The expectaton of e estmato s { ˆ { p p [ p p p p ] 7 ts vaance s a { ˆ a{ / / 8 whee p p, p p, p p p 6 p p p The tue value of e eo-ate to be estmated s p p quaton 7 shows at e estmato s based Howeve, when s lage, e tems w can be gnoed e estmato becomes unbased ate we show at fo e poblems addessed n s epot, s lage, fo ese cases { ˆ p p We use e nomal dstbuton N ˆ { ˆ, a{ ˆ to appoxmate e dstbuton of e estmato Thus e PF of e estmato can be expessed as ˆ a{ ˆ P ˆ ˆ e 9 πa ˆ { We ae nteested n havng e estmated eo-ate be wn a cetan ange of accuacy, say [ ε, ε ], w confdence not less an γ, whee <ε << γ s between Thus, we eque ε ε ˆ a ˆ γ P ˆ ˆ ˆ ˆ d e d πa ˆ ε ε / a ε / a { ˆ { ˆ ε e π t / / a ˆ dt { Q ε whee e functon Q s defned as Q x see Appendx B Changng e fom of, we have Q γ / ε / a ˆ { x e t / / π dt a{ ˆ s a functon of pp, p p, Thus, ncludes sx quanttes, namely ε, γ,, p p,, whee ae unknown ε γ ae gven as pat of e test specfcatons To detemne values fo, whch ae test paametes fo cayng out eo-ate estmaton, some addtonal constants /o objectve functons ae needed Refeng back to ou I test poblem, some quanttes of nteests ae lsted next a nmze x, whch pmaly detemnes e total test tme; b nmze, whch pmaly detemnes e stoage cost fo e coect ones countng; c nmze c xc x, whch s e weghted cost fo bo test tme stoage cost, whee c c ae bo non-negatve cost coeffcent Refeng back to, s e tue eo-ate only known to an oacle, but we can guess a value fo efne ou guess once we have a value fo e estmato The quantty p p s also unknown, but agan we can attempt to appoxmate t Because e appoxmatons ae dffeent fo vaous stuatons, we wll deal w ese ssues n e next secton enttled case studes I CA TUI In [], e numbe of test pattens pe sesson,, has a lowe bound Howeve, ou analyss shows at any postve ntege s feasble fo Fst we consde two cases, namely / e, s vey lage Fom 8 we see at ae nvesely popotonal to each oe o esults n an uppe bound fo, / esults n a lowe bound of Then we consde e symmetc case of p p > At last we consde e geneal case In e followng except n ecton, we assume γ997 hence

6 UC TCHNICA RPORT CNG-8-6 Q γ / { ˆ ε / 9 Then educes to a Case : Fom 8, we have a{ ˆ 9 / 6 / / p p, we have p p p p p p w As p p Fom e defntons of,, we know at all of ese tems ae of e ode of o, 6 ae all of e ode of Keep tems w n e 9 denomnato gnoe tems w hghe odes of, we have a{ˆ can be ewtten as ε / 9 Replacng, w e functons of n leads to ε nce p p ae unknown, we cannot choose e value to be 9 6 / ε Howeve, we can choose to be e maxmum of 9 6 / ε Thus, 5 s satsfed t s guaantees at e estmated eo-ate s n e ange of [ ε, ε ] w confdence γ Fo /, whch s typcally e case, maxmzes 9 ε 6 o we choose to be 6 9 C ε, e, 6 Fo example, fo, ε5, we choose to be 65 When /<<, / maxmzes 9 ε 6 o can be chosen as 6 /, e, 95 ε be 565 Fo, ε5, we choose to Case : / Fo e case of /, not only do we assume at s vey lage, but also at >>, n whch case we gnoe all tems n 8 w an, obtan a{ˆ 7 Ignong e tem w eplacng w ts functon of, 7 educes to a { ˆ / Fom 7, we have 8 ε 8 mla to case, we can choose e value of to be e maxmum of 8 ε nce e maxmum value of 8 ε s 8 ε, we choose as C 8 ε 9 Fo example, fo ε5, we get C 7 us >> 7 Fom 9 we see at when s vey lage, e numbe of test sessons equed n eo-ate estmaton s ndependent of e eo-ate, only depends on e accuacy confdence of e estmaton Ths follows snce, f e eoate s extemely small, en e allowable eo n ou estmaton as specfed by ε allows ou test meodology to wok, fo lage values of eo-ate, ee s enough nfomaton gaeed by usng ese values of C to agan estmate e eo-ate If we choose 7 solve fo usng 9, we get Then e condton to appoxmate 8 w 7 s not satsfed In s case, we cannot use 9 to select Case Case epesent two exteme values of esultng n uppe lowe bounds fo Case : p p > When p p, e pobablty of obsevng a type / o / output ae e same Thus, e expected value of fo each sesson s zeo, e, e sample mean of,, s close to zeo In s case, t appeas at e ones countng test meodology becomes neffectve Howeve, when e estmato 6 educes to / Ths means at e eo-ate can be deved solely fom e vaance of W p p, we have Fom 8, we have a { ˆ

7 UC TCHNICA RPORT CNG-8-7 Assume s lage When compaed w /, / can be gnoed When compaed w /, we can gnoe /, /, / / Thus, educes to { ˆ a / Fom, we have / ε /, e, 9 9 C ε Any postve ntege s legtmate fo When, 9 / ε When s vey lage, 8/ ε These esults fo ae consstent w ose deved fo Cases Fg shows e elatonshp between fo dffeent eo-ates ε 5 ognumbe of Test essons Fom top to bottom, e cuves coespond to e cases of, 5,, Numbe of Test ectos pe esson Fg vs based on q fo dffeent eo-ates ε 5 The ponts maked by an X coespond to e selecton of at mnmze x Consde e case of a sngle stuck at fault n a sngleoutput XOR ccut at causes half of all outputs to be wong Among ose eoneous outputs, half ae of type / e oe half of type / o p p / / Ths povdes an example of Case If e output lne s stuck-at, en agan / but now all eos ae of type / To see how e vaance s nstumental n detemnng e estmated value of eo-ate, agan consde a ccut such at a omly selected half of all possble nput pattens map nto, e oe half map nto Now consde a faulty veson of s ccut, whee agan a omly selected half of all possble nput pattens map nto, e oe half map nto o e tue eo-ate of e faulty ccut s / Now, fo bo e good faulty ccuts, e aveage value of e ones countng s / Thus, e expected value of e dffeence of ones countng between good ccut faulty ccut s zeo In addton, fo e faulty ccut, e pobablty of obsevng a /, /, / o / type esponse s /, e, p p / Assume Fo each sesson, e possble values of ae,,,,,,, If fo a sesson, en all fou outputs ae / type Thus, e pobablty of s / /56 If fo a sesson, en ee outputs ae / type one output s ee / o / type Thus, e pobablty of s x/ x/ / mlaly, we can compute e pobablty of beng,,,,, o As a esult, e pobabltes of beng,,,,,,, ae /56, /, 7/6, 7/, 5/8, 7/, 7/6, / /56, espectvely W e dstbuton functon of, we have e expectaton of to be zeo e vaance to be Fom e expectaton vaance of, we know at e sample mean of,, s about zeo e sample vaance of,, s about Fom e estmato / /, we obtan e estmated eo-ate to be /, whch matches e tue eoate Geneal Case Fo e geneal case, we make no assumptons of e values of p, p γ Howeve, we assume s lage To make ou dscusson clea, we show a copy of 8 below a{ ˆ a{ / / Because s lage, e tem a / can be gnoed when compaed to a / Fo e same eason, e tems /, /, a, a / a / can be gnoed when compaed to / Thus, e vaance of e estmato becomes a{ ˆ w e functons of Replacng,, we have a { ˆ 6 Then, becomes hence 6 ε / Q γ /

8 UC TCHNICA RPORT CNG-8-8 Q γ / 6 5 ε Wout knowng e value of, we choose to be e maxmum achevable value of ght h sde of 5 It can be shown see Appendx C at when s less an /5, whch s geneally e case of nteest, esults n 5 beng maxmal Thus we choose as Q γ / 6 ε Fo γ 997, 6 educes to Ths s expected because Case, fom whch s deved, s a specal case of Case Fo, 6 educes to 6 Fo /, 6 educes to 9 These esults mply at e analyss of e geneal case s consstent w e analyss of ts specal cases ettng allows us to fnd a lowe bound on Fo γ997, s lowe bound s gven by 9 Fo ε, e lowe bound s 8 Usually ε s much smalle an, as ε deceases e lowe bound on nceases Thus n 8, whee an tem exst, t s appopate to gnoe tems contanng,,,, Ths justfes e appoxmatons used to obtan fom 8 Now consde mnmzng test tme whch s popotonal to x Assume γ997 Fom 6, we have 9 When /, en ε 9 6 / ε x has a mnmal value of 9/ / ε When s small, 6/ ε x mn 8 / ε In Fg, e ponts maked by X coespond to e selectons of at mnmze x It can be seen at fo dffeent values of eo-ate, e values of ae almost e same, namely 6/ ε Fo eo-ate estmaton usng sgnatue analyss [], t s ecommended to set /, whch leads to 5/ ε 5 / ε Thus we see at e ones countng technque fo eo-ate estmaton s compaable to sgnatue analyss n tems of total test tme, a lttle hghe n tems of ovehead cost IUATION In e above analyss, we appoxmated e pobablty densty functon of e estmato w e nomal dstbuton N ˆ { ˆ, a{ ˆ Then we developed a way to select to satsfy e accuacy confdence equement of eoate estmaton based on s appoxmaton In s secton, we descbe ou esults of estmatng e eo-ate va smulaton By epeatng e smulaton pocess many tmes, we can collect a lage numbe of estmated eo-ates compae e dstbuton w N ˆ { ˆ, a{ ˆ The smulaton s mplemented as follows A om numbe geneato geneates ee numbes, namely a w pobablty p, w pobablty p w pobablty p p The numbe of s e numbe of s n a sequence of geneated numbes ae counted sepaately The dffeence s ecoded Ths epesents a test sesson test sessons ae caed out, numbes ae ecoded W e estmato pesented n ecton, e eo-ate p p s estmated The above pocedue s epeated tmes esults n eo-ate estmatons The dstbuton of e estmated eo-ates s geneated We use e ATAB tool nomplot to detemne f s data s consstent w a nomal dstbuton Nomplot dsplays e cumulatve dstbuton of e data In e plot, a supemposed lne s dawn to ft e sample data If e data ae nomally dstbuted, e plot appeas lnea Fst, we set p 6, p, 5 Fg a shows e dstbuton of e data t appeas to be stbuton Pobablty ata fom mulaton a Nomal Pobablty Plot 6 8 ata b Fg a stbuton of estmated eo-ate data fom smulaton b The output fom ATAB tool nomplot to test whee e data ae nomally dstbuted Fo s fgue, p 6, p, 5

9 UC TCHNICA RPORT CNG-8-9 nomal The output of nomplot s shown n Fg b, s faly lnea, confmng at e data has a nomal dstbuton Now, a nomal dstbuton can be defned by ts mean stad devaton Fom e data, we estmate e mean to be e vaance to be -6 o e set of eo-ate data has nomal dstbuton N, - In ecton, we used a nomal dstbuton to appoxmate e dstbuton of e estmato The mean of e estmato s gven by 7, esults n a value of The vaance of e estmato s gven by 8, esults n a value of -6 o n ou analyss, we would use N, -6 to be e dstbuton of e estmato Thus ou analytcal esults closely match e smulaton esults Next, we choose dffeent values of whle keepng p 6 p Fg shows e dstbuton of e estmated eo-ate data fom smulaton e nomal dstbuton test fom nomplot fo e case whee s about tmes / Fo ε, e lowe bound on s 8 s close to e lowe bound on, us e case outlned n ecton s satsfed The 5 fgue shows at e smulaton data ae nomally dstbuted Fom e data, we estmate ts nomal dstbuton to be N, 5-7 Fom e analyss n ecton, we obtan e dstbuton of e estmato to be N, 8-7, whch agan s an excellent match Fnally, we consde e case whee p p 5, 5 nce s small, one would expect at fo most test sessons few f any eos would occu In s case, e estmated eo-ate s manly deved on e vaance of e sample data as mentoned n ecton The dstbuton of eo-ate data fom smulaton e output of nomplot ae dsplayed n Fg 5, whch shows e smulaton data has a nomal dstbuton The dstbuton functon of e data s estmated to be N, 985-8, whch matches e dstbuton of e estmato fom analyss, namely N, I CAIFING CHIP IA THIR RROR-RAT One applcaton fo eo-ate estmaton s to assgn chps stbuton 5 stbuton 8 6 Pobablty 6 8 ata fom smulaton a Nomal Pobablty Plot ata b Fg a stbuton of estmated eo-ate data fom smulaton b Output fom ATAB tool nomplot to test whee data ae nomally dstbuted Fo s fgue, p 6, p, Pobablty 6 8 ata fom smulaton a Nomal Pobablty Plot ata b Fg 5 a stbuton of estmated eo-ate data fom smulaton b Output fom ATAB tool nomplot to test whee data ae nomally dstbuted Fo s fgue, p 5, p 5, 5

10 UC TCHNICA RPORT CNG-8- to bns at coespond to eo-ate anges at ae defned by eshold eo-ate values Recall at a eshold sepaates a ange nto two adjacent sub-anges Theshold dvdes e doman of eo-ates nto two ange,,, consequently falng chps ae pattoned nto two types, A B The eo-ate of a type A chp s equal to o less an, whle at of a type B s geate an Testng classfes a chp nto type A o type B accodng to e estmated eoate ˆ Namely, f ˆ <, e chp s classfed as type A; oewse, t s classfed as type B Unfotunately, e om vaable ˆ can be geate an even ough e tue eoate,, s smalle an vce vesa The chance fo s to occu nceases apdly as ˆ appoaches e value o e test can classfy chps ncoectly In statstcs such a test s called a hypoess test In ou case, e two hypoeses ae: H: The chp s type B, e, > ; H: The chp s type A, e, Ths test geneates fou possble outcomes The chp s type B, classfed as type B; e chp s type B, classfed as type A; e chp s type A, classfed as type A; fnally e chp s type A classfed as type B Outcome esults n a lowe pce chp sold eoneously at a hghe pce, whle outcome esults n a hghe pce chp sold at a lowe pce Outcome s lkely acceptable to customes outcome s not o e test should lmt e pobablty of e occuence of outcome Assume t s equed at e pobablty of any type B chp beng classfed as type A be smalle an β, whee β<< Accodng to e analyss of eo-ate estmaton, e estmated eo-ate has nomal dstbuton w ts expectaton beng e tue eo-ate Fg 6 shows e dstbuton of estmated eo-ate of a chp whose tue eoate s a chp whose tue eo-ate s geate an It can be seen at fo a chp w tue eo-ate geate an, e fue e tue eo-ate s fom, e lowe at pobablty of outcome occung Ths pobablty s epesented by e dash aea unde e cuve Howeve, when e tue eo-ate s equal to, e pobablty of outcome s always 5% Thus, e equement s neve satsfed To solve s ssue, we defne anoe eshold, namely n, at s smalle an, postulate e followng classfcaton cteon: a b Fg 6 a PF of estmated eo-ate of a chp whose tue eo-ate s b PF of estmated eo-ate of a chp whose tue eo-ate s geate an If e estmated eo-ate of a chp s smalle an n, t s classfed as type A If e estmated eo-ate of a chp s equal to o geate an n, t s classfed as type B Assume at e pobablty of outcome occung s stll lmted to β Thus s constant also holds when, hence e pobablty of outcome s smalle an β when > o we eque β Pob ˆ < n When, e estmated eo-ate has nomal dstbuton N, a{ˆ Thus, Pob ˆ < n Q n a{ ˆ n Q β β, whch s equvalent to a { ˆ Fom, we aleady have { 6 a ˆ Thus n Q β n 6 7 o equvalently, Q β 6 8 n 8 descbes e equement fo such at e pobablty of outcome occung s lmted to β Wout knowng e value of, we choose to be geate an e maxmum value of e ght h sde of 8 W beng smalle an, e ght h sde of 8 s maxmal at e value Q β when o we n choose accodng to e expesson Q β 9 In 9, β ae gven n e test specfcaton We need to detemne e value of, n Fom ou pevous analyss, e selecton of depends on what cost functon s mnmzed Assume has been detemned Then s detemned by n As n deceases, deceases Because smalle values of esult n less stoage cost, t s mpotant to keep small Howeve, smalle n causes moe type A chps to be classfed as type B, meanng a loss n poft o ee s a tadeoff between stoage cost poft Fo a type A chp, ts tue eo-ate,, may be smalle o geate an n Gven, e pobablty of outcome can be computed Consde e case of < n The estmated eo-ate has nomal dstbuton 6 N The pobablty of,

11 UC TCHNICA RPORT CNG-8- outcome, e e estmated eo-ate beng geate an n s n Q / 6 When <, takes on a maxmal value of n Q, whch s e uppe bound of / e pobablty of outcome fo e case of < n When n < <, e pobablty of outcome s n Q / 6 nce p p,, t can be shown at f < / en s maxmal when has e value n Q / 5 6 mlaly, e pobablty of outcome fo e case of > can be computed Table summazes e uppe bounds on e pobablty of eoneous classfcaton fo ese ee cases mla to e eo-ate estmaton technque based on sgnatue analyss [], t s not necessay to apply all test sessons befoe makng a decson because e fae e tue eo-ate away fom e eshold, e less pobablty of makng wong classfcaton The numbe of test sessons s based on e assumpton of e wost case, e, when Howeve, a defectve chp usually does not epesent e wost case, sometmes neve epesents such a case The classfcaton equement s at e pobablty of an estmated eo-ate less an n be smalle an β f e tue eo-ate s geate an o t s possble to make a decson wout executng all test sessons as long as e pobablty of makng a wong decson s smalle an β et ms be e mnmal numbe of test sessons equed fo a chp w eo-ate at satsfes e constant mposed by β Fo >, e uppe bound on e pobablty at e estmated eo-ate s smalle an n s n Q Fo < n, e uppe bound of / e pobablty at estmated eo-ate s geate an n s n Q Bo fomulas ae lsted n / Table Then ms should satsfy n Q β f >, / ms ms n Q β f < n / ms ms TAB I TH AXIA IKIHOO OF AKING AN RRONOU CAIFICATION Tue eo-ate < n, estmated eo-ate ˆ > n Type A classfed as type B n << ˆ > n Type A classfed as type B ˆ < n > Type B classfed as type A Uppe Bound of e Pobablty of akng oneous Classfcaton Q Q n / When n < <, ms s equal to as specfed by 9 As an example, let n 9, β 5 s chosen to be / Fg 7 shows e mnmal numbe of test sessons ms fo dffeent tue eo-ates As moves away fom e ange of [ n, ], ms quckly deceases o fo eo-ates fa fom [ n, ], only a small numbe of test sessons ae needed To be able to make ealy decson befoe applyng all test sessons, e test pocedue must be modfed An ognal test s dvded nto multple phases, each of whch conssts of a dsjont subset of e test sessons Afte each test phase s completed, e eo-ate s estmated based on e esults fom e test sessons appled so fa, assumng e estmated eo-ate s e tue eo-ate If e estmated eo-ate s smalle an n, e pobablty of makng an eoneous classfcaton s calculated usng If e estmated eo-ate s geate an, e pobablty of makng an eoneous classfcaton s calculated usng If e computed pobablty s smalle an β, e test stops, oewse t contnues If e estmated eo-ate s n e ange [ n, ], e test also contnues unless all test sessons nmal numbe of test sessons, ms n 5 / n Q 6 5 Tue eo-ate, Fg7 The mnmal numbe of test sessons fo dffeent tue eo-ates

12 UC TCHNICA RPORT CNG-8- have been un, n whch case testng s fnshed II XPRINTA RUT To valdate s chp classfcaton technque, we appled t to e ICA 85 benchmak ccut C at has 7 pmay outputs Because ou technque s cuently only applcable to sngle output ccuts, we ceated seven sngle output ccuts fom C, labeled as C_, C_, C_, C_, C_5, C_6 C_7 The seven ccuts have e same netlst as C Fo each of ese netlsts, only one pmay output of C s teated as e output of e new ccut oe outputs ae teated as ntenal wes Fo example, e output pn of C s e output of C_, output pn,,, 5, 6 7 of C ae teated as ntenal wes n C_ To model a defect, we used e sngle stuck-at fault model ach of ese ccuts has 86 sngle stuck-at faults Thus, coespondng to each fault-fee ccut, ee ae 86 faulty ccuts Consde e 86 faulty ccuts of C_7 nce we know e actual faults n e ccut, we can obtan e actual eo-ates see Fg 8 To classfy ese ccuts, we set, n 9, β5 5 The eo-ate of a type A ccut s n e ange,, fo a type B ccut [, Fom 9, e maxmal numbe of test sessons,, s 8 We use e 5 mult-phase test scheme descbed n e pevous secton fo classfcaton In e fst phase, * test sessons ae executed The collected data s not statstcally meanngful f * s too small In e followng phases, only one test sesson s appled ven ough moe test sessons can be appled, we choose one because we want to fnd e exact numbe of e test sessons needed Afte each phase, e estmated eo-ate s computed e condton fo stoppng s checked The total numbe of test sessons fo each ccut s ecoded Fg 9 shows e hstogam of test sessons fo all e ccuts Fom Fg 9, t s seen at fo many ccuts, only a small numbe of test sessons ae needed Ths means at e eoate fo each of ese ccuts s fa away fom e ange 9, Ths esult s consstent w Fg 8, at shows at only a small facton of e ccuts have an eo-ate among nea e ange 9, To fue demonstate e coelaton between e numbe of test sessons eo-ate, n Table we lst e aveage numbe of test sessons fo dffeent sub-anges of eo-ate Fo eo-ate fa fom 9~, e aveage numbe of test sessons s small As e eo-ate gets close to 9~, e aveage numbe of test sessons nceases Ths s consstent w e analyss n ecton 6 Fo eo-ates n e ange 9~, e aveage numbe of test sessons s 995, whch s not equal to e maxmal numbe of test 8 stbuton 5 stbuton 6 stbuton o-ate a o-ate 5 b Fg8 Nomalzed hstogam of eo-ates of 86 faulty ccuts assocated w C_7 a The eo-ate ange s fom to b The eo-ate ange s fom to 55 Note: e scales of a b ae dffeent stbuton Test essons a 6 8 Test essons b Fg9 Nomalzed hstogam of e numbe of test sessons appled when classfyng e 86 faulty ccuts of C_7 a The numbe of test sessons s n e ange of to b The numbe of test sessons s n e ange of to Note: e scales of a b ae dffeent

13 UC TCHNICA RPORT CNG-8- sessons 8 Ths s because fo some ccuts, e estmated eo-ate s ee below 9 o above, stoppng condton s satsfed befoe unnng 8 test sessons TAB II ARAG NUBR OF TT ION OF FAUT C_7 CIRCUIT FOR IFFRNT RROR-RAT RANG o-ate Range The pecentage of ccuts msclassfed as type A s called test escape, ose msclassfcaton as type B s called yeld loss Intutvely, f e eo-ate of a ccut s fa fom e eshold, t s less lkely to be msclassfed Because most of e 86 ccuts n ou expement have an eo-ate fa fom e eshold, e test escape yeld loss should be low Table lsts e numbe of msclassfcaton fo ee dffeent eo-ate anges Thus fo s expement e test escape s /7% e yeld loss s /9% The same classfcaton expement was appled to benchmak ccut C88 A sngle output ccut, namely C88_6, was deved fom C88 such at one pmay output of C88 s e only output of C88_6, e oe pmay outputs of C88 ae teated as ntenal wes of C88_6 Because C88_6 has 96 sngle stuck-at faults, we obtaned 96 faulty copes of C88_6 ach faulty copy coesponds to a sngle stuck-at fault In e expement,, n 9, β5 5 Table lsts e aveage numbe of test sessons fo dffeent sub-anges of eo-ate e numbe of faulty ccuts n each ange mla to e esults fo C_7, e numbe of test sessons nceases when e eo-ate ange s close to 9~ Table also quantfes e numbe of msclassfcatons made Of e 55 type A ccuts, 7 ae msclassfed Of e 6 type B ccuts, 6 ae msclassfed The esultng yeld loss s 5% e test escape s 9% The esults of ese expements on C_7 C88_6 ae consstent w ou analytcal esults When e eo-ate of a ccut s fa fom e eshold, coect classfcaton o-ate Range Aveage Numbe of Test essons TAB III ICAIFICATION IN TH XPRINT FOR C_7 Actual Type Actual Numbe of Faulty Ccuts Numbe of Ccuts n e Range ~ 77 ~8 98 8~ 8 7 ~ ~7 9 7~ ~ ~ 5 9 ~ 59 8 ~ ~ 56 ~5 99 5~ 5 7 Numbe of sclassfed Ccuts ~9 Type A 55 9~ Type A 6 9 ~ Type B 9 decson can be made afte a small numbe of test sessons As e eo-ate of a ccut appoaches e eshold, moe test sessons ae needed In ou expements, we choose β to be 5, whch lmts e pobablty of msclassfcaton, use β fo e wose stuaton to detemne oe test paametes When e eo-ate of a ccut s fa fom e eshold, e pobablty of makng a wong decson s actually smalle an β Fom table t s seen at e pobablty of makng a wong decson nceases as e actual eo-ate appoaches e eshold o-ate Range TAB I XPRINTA RUT FOR FAUT C88_6 CIRCUIT Actual Type Aveage Numbe of Test essons Numbe of Actual Ccuts n e Range Numbe of sclassfed Ccuts ~ A 97 ~8 A ~ A 6 7 ~5 A ~7 A 7 7 7~9 A ~ A ~ B ~ B 7 ~7 B ~ B 68 7 ~5 B ~ B 9 9 In e expement fo C88_6, 57 type A ccuts ae coectly classfed as type A, 6 type B ccuts ae msclassfed as type A Thus, 55 ccuts ae classfed as type A w 6 of em actually beng type B If we sell all 55 ccuts, n pncple 6 mght justfable be etuned The facton of chps at ae beyond-toleance FBT, namely 6/55, should not be confused w e classcal noton of defects pe mllon P used as a measue fo poduct qualty We eque at e maxmal pobablty of msclassfcaton fo a ccut be β Theoetcally, fo e wost scenao n classfcaton, e numbe of actual type B ccuts at ae msclassfed as type A s β e numbe of actual type B ccuts; e numbe of actually type A ccuts at ae msclassfed as type B s β e numbe of actual type A ccuts Thus e numbe of ccuts at ae classfed as type A s e numbe of actual type A ccuts β e numbe of actual type A ccuts β e numbe of actual type B ccuts Then, FBT s equal to β e numbe of actual type B ccuts e numbe of ccuts at ae classfed as type A In pactce, e pobablty of msclassfcaton s lowe an β when e eo-ate of a ccut s fa fom e eshold Thus, e eal value of FBT s usually lowe en ts eoetcal value In ou expement on e 96 faulty ccuts of C88_6, e expemental value of FBT s 6/558%, whle e eoetcal value of FBT s 5x6/55-5x555x6 o 696%

14 UC TCHNICA RPORT CNG-8- III CONCUION o-toleance mates new domans fo I tests, such as eo-ate Hee, e objectve s to classfy chps accodng to e eo-ate ae an pass/fal, o X Hz vs Hz The fundamental task assocated w s classfcaton s e estmaton of eo-ate An eo-ate estmaton technque based on sgnatue analyss was pesented n [] [], whee t was shown at s estmaton chp classfcaton can be effectvely mplemented w easonable BIT test esouce Anoe eo-ate estmaton technque based on ones countng also esulted n an estmato classfcaton technque [] The statstcal popetes of s estmato wee not fully studed, some esults wee non-ntutve, e equed esouces needed to cay out e test pocess appeaed to be lage an necessay In s epot, we agan consde e poblem addessed n [], develop a maematcally smple estmato In addton, we analyze e statstcal chaactestcs of e estmato, such as ts expectaton, vaance pobablty densty functon We detemne e condtons when s estmato s based vs unbased Based on e statstcal analyss, we descbe a meod fo selectng e values fo two key test paametes, namely e numbe of test pattens pe sesson,, e numbe of test sessons, These paametes mpact e test esouces used fo eo-ate estmaton We show how ese paametes ae a functon of bo test tme stoage equements In addton, we povde useful pactcal bounds fo ese two paametes The esults of ou analyss show at e test esouces equed fo eo-ate estmaton based on ones countng ae qute compaable to ose needed when sgnatue analyss s employed Thus, e majo dffeence n ese two appoaches s at sgnatue based eo-ate analyss can opeate on a mult-output ccut, whle ones countng pocesses one output at a tme unless e test hadwae s eplcated fo each output lne We also addess e poblem of classfyng chps based on e eo-ate The poposed classfcaton pocedue s pattoned nto multple phases Ths pocess can sgnfcantly educe classfcaton tme wout any loss n e qualty of classfcaton Thoughout s epot seveal assumptons have been made, such as assumng a nomal dstbuton fo some om vaables, doppng tems at lead to second o d ode effects We have addessed ese ssues n ou expements, valdated at ese assumptons ae appopate We have also consdeed vaous bounday o exteme condtons, such as whee e test leng of a test sesson s o vey lage In ou analyss, we assume all possble nput pattens ae equally lkely to appea have unfom dstbuton Thus e test patten geneato n BIT s mplemented w FR geneates pseudo om test pattens Fo some ccuts, e functonal nput may have dffeent statstcs an unfom dstbuton To deal w s stuaton, test pattens can be obtaned fom functonal test ccut nstead of FR, e same pocedue s appled Thus, e estmated eoate s a condtonal pobablty of outputtng an eo gven e eal statstcs of nput pattens Thus, f e dstbuton of functonal nput pattens s known, t may be feasble to eplace e PRPG w one at poduces pattens whose dstbuton moe closely mmcs e eal wold Fo multple output ccuts, e output lnes may not be of e same mpotance Thus, when ese output lnes ae consdeed ndependently, each of ese lnes wll be assocated w a dffeent eo-ate eshold Fo example, e most sgnfcance bt lne mght have a much smalle eoate eshold an e least sgnfcance bt lne Unde s stuaton, e technque pesented n s epot can be used to estmate e eo-ate of each output lne justfy whee e eo-ate s smalle an a eshold When e output lnes of a multple output ccut ae consdeed as a whole fo eo-ate, ts eo-ate cannot be smply deved fom e eo-ate of each output lne because of e coelaton of eos among dffeent output lnes Ths poblem has been solved by usng sgnatue analyss [], whle usng ones countng to solve s poblem s e bass of a focomng pape APPNIX A xpectaton aance of e stmato Fo eo-ate estmaton based on ones countng compesson, we popose usng e estmato ˆ, whee ae, espectvely, e sample mean sample vaance of e sampled om vaable In s appendx, we deve expessons fo e expectaton vaance of e estmato Pelmnaes In ecton, we defned X, whee X, X,, X ae d have e same dstbuton as e om vaable X The PF of X s PobXp, PobX p, PobXp p, whee p, p p p To smplfy futue maematcal expessons, we defne,, as follows: X p, { X X p { p p p p p { { X X p, X p p p p p p p, p

15 UC TCHNICA RPORT CNG-8-5 { X { X X X X X p p 6 p p s e expectaton of X,, ae e nd, d cental moments of X Addtonal symbols ae defned below {,,,,, { {, { {, { {, { { denotes e expectaton of,,, ae d have e same dstbuton as,, ae e st, nd, d ode moments of nce X n tems of,, { { X { a, we ae able to expess,, { { p p { X X X { { X { X { X X X { X { { X X { evaton of e xpectaton aance of e stmato The expectaton of e sample vaance of a om vaable s equal to e expectaton of e om vaable [] o we have { a{ A The expectaton of s computed as follows { j j, j { { { { { a{ { A Thus, e expectaton of e estmato s { { { a{ a{ { / / / p p [ p p p p ] A Next we compute e vaance of e estmato a a a Cov, a{ {, a Cov A We next compute each component n A Now { { { a A5 whee { a{ { { A6 Thus, { 6 a A7 a et We next consde e tem { / Then

16 UC TCHNICA RPORT CNG-8-6 Recall at { { { a { { { a A8 Hence, { { { A9 In A9, { A { { A { { A Theefoe, { A o fnally fo A8 we have { { a A We now consde e tem { { {, Cov n A, whee { A5 { { { { a A6 Theefoe,, Cov A7 ubsttutng { a, { a, Cov nto A usng A7, A A7, e vaance of e estmato s obtaned as a A8 Because s lage compaed to, we assume at Now we have a A9 B Q Functon: Qx Q functon Qx computes e ght tal pobablty of nomal dstbuton N, o Q functon s defned as x t dt e x Q / π Usng Q functon, many pobablty elated to a om vaable havng nomal dstbuton N, σ can be smply expessed smply Hee ae some examples et T denote such a om vaable Pobablty of T beng n x, s x t dt e x Q / / σ πσ σ Pobablty of T beng n -, x s - Qx-/σ Pobablty of T beng n -,, whee >, s - Q/σ C Justfcaton of 5 w </5 beng axmzed when n ecton quaton 5 n ecton states at Q 6 / ε γ

17 UC TCHNICA RPORT CNG-8-7 Because p p p p, we have < < Thus s maxmzed when o s The tem 6 T can be ewtten as a 5 / 6 7 / 6 T 6 C When > 5 / 6, T deceases as nceases Fo </5, 5 / 6< On e oe h, s always nonnegatve Fo </5, T s maxmzed when Thus, fo </5, bo 5 / 6 7 / 6 6 a ae maxmzed when Hence, fo </5, Q γ / 6 ε s maxmzed when, ts maxmum s Q γ / ε C [] Z Pan A Beue, stmatng o Rate n efectve ogc Usng gnatue Analyss, I Tans on Computes, ol 56, No 5, pp 65-66, ay 7 [] hahd K Gupta, stmatng o Rate ung elf-test va One s Countng, Poc Int l Test Conf, pape 5, 6 [] G Casella R Bege, tatstcal Infeence, uxbuy Pess, nd ed, [5] A Beue H Zhu, o-toleance ult-eda, Poc Int l Conf on Intellgent Info Hdng ultmeda, pp 5-5, 6 [6] Abamovc, A Beue A Fedman, gtal ystems Testng Testable esgn, Wley-I Pess, 99 [7] N K Jha K Gupta, Testng of gtal ystems, Cambdge Unv Pess, st ed, ACKNOWGNT The auos ank hdeh hahd Pofessos eep Gupta, Ke Chugg Antono Otega fo e valuable suggestons on s wok ome deas mpovements n s epot ae attbutable to em RFRNC [] Intenatonal Technology Roadmap fo emconductos: eld nhancement, 6 Update, [] Butts, A ehon C Goldsten, olecula lectoncs: evces, ystems Tools fo Ggagate, Ggabt Chps, Poc Int l Conf on Compute Aded esgn, pp -, Nov -, [] R I Baha, Tends Futue ectons n Nano tuctue Based Computng Fabcaton, Poc Int l Conf on Compute esgn, Oct -, 6 [] I Koen Z Koen, efect Toleance n I Ccuts: Technques eld Analyss, Poc I, ol 86, No 9, pp 89-88, ept 998 [5] A Beue, K Gupta T ak, efect o Toleance n e Pesence of assve Numbes of efects, I esgn Test of Computes, pp 6-7, ay-june [6] A Beue, Intellgble Test Technques to uppot o-toleance, Poc Asan Test ymp, pp 86-9, [7] H H Kuok, Audo Recodng Appaatus Usng an Impefect emoy Ccut, U Patent 5758, Patent Tademak Offce, 995; [8] P Agade, gtal ecetay, U patent 56555, Patent Tademak Offce, 997; [9] H Chung A Otega, Analyss Testng fo o Toleant oton stmaton, Poc I Int l ymp on efect Fault Toleance n I ystems, pp 5-5, Nov 5 [] H Chung A Otega, ystem evel Fault Toleance fo oton stmaton, Techncal Repot UC-IPI5, gnal Image Pocessng Insttute, Unv of ouen Calfona, [] A Beue, stmatng o Rate n o Toleant I Chps, I Int l Wokshop on lectonc esgn, Test Applcatons, pp - 6, Januay